I have never like "Laplace Transform"- too mechanical. In particular, for this problem, using the Laplace Transform is using a cannon to kill a fly.

Here is how I would do this problem: differentiate both sides of the equation, with respect to t, to get rid of the integral: y(t)- y''(t)= 1 or y''- y= -1. We have initial conditions y(0)= 0 and y'(0)= 0 (from setting t=0 in the orginal equation). That's a very simple linear equation with constant coefficients. The characteristic equation has real roots so the solution will involve exponentials (and so cosh as haruspex says) not trig functions.